Propositional Satisfiability Problem Research Articles

Data-Aware Declarative Process Mining with SAT

Process Mining is a family of techniques for analyzing business process execution data recorded in event logs. Process models can be obtained as output of automated process discovery techniques or can be used as input of techniques for conformance checking or model enhancement. In Declarative Process Mining, process models are represented as sets of temporal constraints (instead of procedural descriptions where all control-flow details are explicitly modeled). An open research direction in Declarative Process Mining is whether multi-perspective specifications can be supported, i.e., specifications that not only describe the process behavior from the control-flow point of view, but also from other perspectives like data or time. In this article, we address this question by considering SAT (Propositional Satisfiability Problem) as a solving technology for a number of classical problems in Declarative Process Mining, namely, log generation, conformance checking, and temporal query checking. To do so, we first express each problem as a suitable FO (First-Order) theory whose bounded models represent solutions to the problem, and then find a bounded model of such theory by compilation into SAT.

Initial Solution Generation and Diversified Variable Picking in Local Search for (Weighted) Partial MaxSAT.

The (weighted) partial maximum satisfiability ((W)PMS) problem is an important generalization of the classic problem of propositional (Boolean) satisfiability with a wide range of real-world applications. In this paper, we propose an initialization and a diversification strategy to improve local search for the (W)PMS problem. Our initialization strategy is based on a novel definition of variables' structural entropy, and it aims to generate a solution that is close to a high-quality feasible one. Then, our diversification strategy picks a variable in two possible ways, depending on a parameter: continuing to pick variables with the best benefits or focusing on a clause with the greatest penalty and then selecting variables probabilistically. Based on these strategies, we developed a local search solver dubbed ImSATLike, as well as a hybrid solver ImSATLike-TT, and experimental results on (weighted) partial MaxSAT instances in recent MaxSAT Evaluations show that they outperform or have nearly the same performances as state-of-the-art local search and hybrid competitors, respectively, in general. Furthermore, we carried out experiments to confirm the individual impacts of each proposed strategy.

A cerebellar operant conditioning-inspired constraint satisfaction approach for product design concept generation

Conceptual design is a pivotal stage of new product development. The function-behaviour-structure framework is adopted in this stage to help designers search design space and generate conceptual solutions iteratively. Computer-aided methods developed within this framework will yield significant insight into facilitating the cognitive activities of designers. In order to solve the mapping process from behaviours to structures which is a typical constraint satisfaction problem, a cerebellar operant conditioning-inspired constraint satisfaction approach is proposed in this paper. The design constraints-driven operant conditioning and its regulation mechanism by the cerebellum are analysed for the first time. Proposition logic is applied to transfer the constraint satisfaction problem into a propositional satisfiability problem while an undirected graph is utilised to model design space. Inspired by the modularised cerebellar structure, a modularised constraint satisfaction neural network is constructed to determine the satisfiability of design problems. Conceptual solutions can be generated by clustering the embedding of nodes in this network. The proposed approach imitates the design constraint-driven operant conditioning to narrow down design space without assigning specific values to design components. It reduces design iterations and avoids combinatorial explosions during conceptual design.

A Time-Efficient and Exploratory Algorithm for the Rectangle Packing Problem

Today, resource waste is considered as one of the most important challenges in different industries. In this regard, the Rectangle Packing Problem (RPP) can affect noticeably both time and design issues in businesses. In this study, the main objective is to create a set of non-overlapping rectangles so that they have specific dimensions within a rectangular plate with a specified width and an unlimited height. The ensued challenge is an NP-complete problem. NP-complete problem, any of a class of computational problems that still there are no efficient solution for them. Most substantial computer-science problems such as the traveling salesman problem, satisfiability problems (sometimes called propositional satisfiability problem and abbreviated SAT or B-SAT), and graph-covering problems are belong to this class. Essentially, it is complicated to spot the best arrangement with the highest rate of resource utilization by emphasizing the linear computation time. This study introduces a time-efficient and exploratory algorithm for the RPP, including the lowest front-line strategy and a Best-Fit algorithm. The obtained results confirmed that the proposed algorithm can lead to a good performance with simplicity and time efficiency. Our evaluation shows that the proposed model with utilization rate about 94.37% outperforms others with 87.75%, 50.54%, and 87.17% utilization rate, respectively. Consequently, the proposed method is capable to of achieving much better utilization rate in comparison with other mentioned algorithms in just 0.023 s running-time, which is much faster than others.

Focused random walk with probability distribution for SAT with long clauses

Focused random walk (FRW) is one of the most influential paradigm of stochastic local search (SLS) algorithms for the propositional satisfiability (SAT) problem. Recently, an interesting probability distribution (PD) strategy for variable selection was proposed and has been successfully used to improve SLS algorithms, resulting in state-of-the-art solvers. However, most solvers based on the PD strategy only use polynomial function (PoF) to handle the exponential decay and are still unsatisfactory in dealing with medium and huge k-SAT instances at and near the phase transition. The present paper is focused on handling all k-SAT instances with long clauses. Firstly, an extensive empirical study of one state-of-the-art FRW solver WalkSATlm on a wide range of SAT problems is presented with the focus given on fitting the distribution of the break value of variable selected in each step, which turns out to be a Boltzmann function. Using theses case studies as a basis, we propose a pseudo normal function (PNF) to fit the distribution of the break value of variable selected, which is actually a variation of the Boltzmann function. In addition, a new tie-breaking flipping (TBF) strategy is proposed to prevent the same variable from being flipped in consecutive steps. The PNF based PD strategy combined with the TBF strategy lead to a new variable selection heuristic named PNF-TBF. The PNF-TBF heuristic along with a variable allocation value (Vav) function are used to significantly improve ProbSAT, a state-of-the-art SLS solver, leads to a new FRW algorithm dubbed PNFSat, which achieves the state-of-the-art performance on a broad range of huge random 7-SAT instance near the phase transition as demonstrated via the extensive experimental studies. Some further improved versions on top of PNFSat are presented respectively, including PNFSat_alt, which achieves the state-of-the-art performance on the medium 7-SAT instances at the phase transition; PNP as well as an integrated version of these three algorithms, named PDSat, which achieves the state-of-the-art performances on all huge and medium random k-SAT instances with long clauses as demonstrated via the comparative studies using different benchmarks.

ON OBDD-BASED ALGORITHMS AND PROOF SYSTEMS THAT DYNAMICALLY CHANGE THE ORDER OF VARIABLES

AbstractIn 2004 Atserias, Kolaitis, and Vardi proposed $\text {OBDD}$ -based propositional proof systems that prove unsatisfiability of a CNF formula by deduction of an identically false $\text {OBDD}$ from $\text {OBDD}$ s representing clauses of the initial formula. All $\text {OBDD}$ s in such proofs have the same order of variables. We initiate the study of $\text {OBDD}$ based proof systems that additionally contain a rule that allows changing the order in $\text {OBDD}$ s. At first we consider a proof system $\text {OBDD}(\land , \text{reordering})$ that uses the conjunction (join) rule and the rule that allows changing the order. We exponentially separate this proof system from $\text {OBDD}(\land )$ proof system that uses only the conjunction rule. We prove exponential lower bounds on the size of $\text {OBDD}(\land , \text{reordering})$ refutations of Tseitin formulas and the pigeonhole principle. The first lower bound was previously unknown even for $\text {OBDD}(\land )$ proofs and the second one extends the result of Tveretina et al. from $\text {OBDD}(\land )$ to $\text {OBDD}(\land , \text{reordering})$ .In 2001 Aguirre and Vardi proposed an approach to the propositional satisfiability problem based on $\text {OBDD}$ s and symbolic quantifier elimination (we denote algorithms based on this approach as $\text {OBDD}(\land , \exists )$ algorithms). We augment these algorithms with the operation of reordering of variables and call the new scheme $\text {OBDD}(\land , \exists , \text{reordering})$ algorithms. We notice that there exists an $\text {OBDD}(\land , \exists )$ algorithm that solves satisfiable and unsatisfiable Tseitin formulas in polynomial time (a standard example of a hard system of linear equations over $\mathbb {F}_2$ ), but we show that there are formulas representing systems of linear equations over $\mathbb {F}_2$ that are hard for $\text {OBDD}(\land , \exists , \text{reordering})$ algorithms. Our hard instances are satisfiable formulas representing systems of linear equations over $\mathbb {F}_2$ that correspond to checksum matrices of error correcting codes.

A Hybrid Approach to Solve SAT and UNSAT Problems

This work involves the construction of a hybrid approach to solve SAT and UNSAT problems. The solution combines techniques found in Stalmarck and DPLL algorithms for solving propositional satisfiability problems. We extend the Stalmarck rules in order to reduce computational cost during the deduction phase. We applied our solution to SAT and UNSAT instances. From the results it was found that this approach showed good results for UNSAT industrial instances. In some cases, our approach obtained gains of 45% to 60% in terms of efficient when compared to existing approaches. We compared our solution against the best known SAT solvers such as ZChaff and RSat. The techniques and methods involved are described in the article.

Solving complex problems using model transformations: from set constraint modeling to SAT instance solving

On the one hand, solvers for the propositional satisfiability problem (SAT) can deal with huge instances composed of millions of variables and clauses. On the other hand, Constraint Satisfaction Problems (CSP) can model problems as constraints over a set of variables with non-empty domains. They require combinatorial search methods as well as heuristics to be solved in a reasonable time.In this article, we present a technique that benefits from both expressive CSP modeling and efficient SAT solving. We model problems as CSP set constraints. Then, a propagation algorithm reduces the domains of variables by removing values that cannot participate in any valid assignment. The reduced CSP set constraints are transformed into a set of suitable SAT instances. They may be simplified by a preprocessing method before applying a standard SAT solver for computing their solutions.The practical usefulness of this technique is illustrated with two well-known problems: a) the Social Golfer, and b) the Sports Tournament Scheduling. We obtained competitive results either compared with ad hoc solvers or with hand-written SAT instances. Compared with direct SAT modeling, the proposed technique offers higher expressiveness, is less error-prone, and is relatively simpler to apply. The automatically generated propositional satisfiability instances are rather small in terms of clauses and variables. Hence, applying the constraint propagation phase, even huge instances of our problems can be tackled and efficiently solved.

Enhanced Membrane Computing Algorithm for SAT Problems Based on the Splitting Rule

Boolean propositional satisfiability (SAT) problem is one of the most widely studied NP-complete problems and plays an outstanding role in many domains. Membrane computing is a branch of natural computing which has been proven to solve NP problems in polynomial time with a parallel compute mode. This paper proposes a new algorithm for SAT problem which combines the traditional membrane computing algorithm of SAT problem with a classic simplification rule, the splitting rule, which can divide a clause set into two axisymmetric subsets, deal with them respectively and simultaneously, and obtain the solution of the original clause set with the symmetry of their solutions. The new algorithm is shown to be able to reduce the space complexity by distributing clauses with the splitting rule repeatedly, and also reduce both time and space complexity by executing one-literal rule and pure-literal rule as many times as possible.

Identifying Bottlenecks in Practical SAT-Based Model Finding for First-Order Logic Ontologies with Datasets

Satisfiability of first-order logic (FOL) ontologies is typically verified by translation to propositional satisfiability (SAT) problems, which is then tackled by a SAT solver. Unfortunately, SAT solvers often experience scalability issues when reasoning with FOL ontologies and even moderately sized datasets. While SAT solvers have been found to capably handle complex axiomatizations, finding models of datasets gets considerably more complex and time-intensive as the number of clause exponentially increases with increase in individuals and axiomatic complexity. We identify FOL definitions as a specific bottleneck and demonstrate via experiments that the presence of many defined terms of the highest arity significantly slows down model finding. We also show that removing optional definitions and substituting these terms by their definiens leads to a reduction in the number of clauses, which makes SAT-based model finding practical for over 100 individuals in a FOL theory.

Computing and Explaining Query Answers over Inconsistent DL-Lite Knowledge Bases


 
 
 Several inconsistency-tolerant semantics have been introduced for querying inconsistent description logic knowledge bases. The first contribution of this paper is a practical approach for computing the query answers under three well-known such semantics, namely the AR, IAR and brave semantics, in the lightweight description logic DL-LiteR. We show that query answering under the intractable AR semantics can be performed efficiently by using IAR and brave semantics as tractable approximations and encoding the AR entailment problem as a propositional satisfiability (SAT) problem. The second issue tackled in this work is explaining why a tuple is a (non-)answer to a query under these semantics. We define explanations for positive and negative answers under the brave, AR and IAR semantics. We then study the computational properties of explanations in DL-LiteR. For each type of explanation, we analyze the data complexity of recognizing (preferred) explanations and deciding if a given assertion is relevant or necessary. We establish tight connections between intractable explanation problems and variants of SAT, enabling us to generate explanations by exploiting solvers for Boolean satisfaction and optimization problems. Finally, we empirically study the efficiency of our query answering and explanation framework using a benchmark we built upon the well-established LUBM benchmark.
 
 

On the Parameterized Complexity of Finding Small Unsatisfiable Subsets of CNF Formulas and CSP Instances

In many practical settings it is useful to find a small unsatisfiable subset of a given unsatisfiable set of constraints. We study this problem from a parameterized complexity perspective, taking the size of the unsatisfiable subset as the natural parameter where the set of constraints is either (i) given a set of clauses, i.e., a formula in conjunctive normal Form (CNF), or (ii) as an instance of the Constraint Satisfaction Problem (CSP). In general, the problem is fixed-parameter in tractable. For an instance of the propositional satisfiability problem (SAT), it was known to be W[1]-complete. We establish A[2]-completeness for CSP instances, where A[2]-hardness prevails already for the Boolean case. With these fixed-parameter intractability results for the general case in mind, we consider various restricted classes of inputs and draw a detailed complexity landscape. It turns out that often Boolean CSP and CNF formulas behave similarly, but we also identify notable exceptions to this rule. The main part of this article is dedicated to classes of inputs that are induced by Boolean constraint languages that Schaefer [1978] identified as the maximal constraint languages with a tractable satisfiability problem. We show that for the CSP setting, the problem of finding small unsatisfiable subsets remains fixed-parameter intractable for all Schaefer languages for which the problem is non-trivial. We show that this is also the case for CNF formulas with the exception of the class of bijunctive (Krom) formulas, which allows for an identification of a small unsatisfiable subset in polynomial time. In addition, we consider various restricted classes of inputs with bounds on the maximum number of times that a variable occurs (the degree), bounds on the arity of constraints, and bounds on the domain size. For the case of CNF formulas, we show that restricting the degree is enough to obtain fixed-parameter tractability, whereas for the case of CSP instances, one needs to restrict the degree, the arity, and the domain size simultaneously to establish fixed-parameter tractability. Finally, we relate the problem of finding small unsatisfiable subsets of a set of constraints to the problem of identifying whether a given variable-value assignment is entailed or forbidden already by a small subset of constraints. Moreover, we use the connection between the two problems to establish similar parameterized complexity results also for the latter problem.

Implementing Efficient All Solutions SAT Solvers

All solutions SAT (AllSAT for short) is a variant of the propositional satisfiability problem. AllSAT has been relatively unexplored compared to other variants despite its significance. We thus survey and discuss major techniques of AllSAT solvers. We accurately implemented them and conducted comprehensive experiments using a large number of instances and various types of solvers including a few publicly available software. The experiments revealed the solvers’ characteristics. We made our implemented solvers publicly available so that other researchers can easily develop their solvers by modifying our code and comparing it with existing methods.

The Configurable SAT Solver Challenge (CSSC)

It is well known that different solution strategies work well for different types of instances of hard combinatorial problems. As a consequence, most solvers for the propositional satisfiability problem (SAT) expose parameters that allow them to be customized to a particular family of instances. In the international SAT competition series, these parameters are ignored: solvers are run using a single default parameter setting (supplied by the authors) for all benchmark instances in a given track. While this competition format rewards solvers with robust default settings, it does not reflect the situation faced by a practitioner who only cares about performance on one particular application and can invest some time into tuning solver parameters for this application. The new Configurable SAT Solver Competition (CSSC) compares solvers in this latter setting, scoring each solver by the performance it achieved after a fully automated configuration step. This article describes the CSSC in more detail, and reports the results obtained in its two instantiations so far, CSSC 2013 and 2014.

DPLL算法中的变量决策启发式策略

针对命题公式φ(CNF形式)的可满足性问题的求解效率问题，基于对DPLL完全算法的学习和在不同形式下对子句文字排序处理的研究，提出了一种新的求解SAT问题的算法。新算法根据子句长度对命题公式φ进行分组，并根据变量出现次数进行初始排序，然后考虑变量中正负文字出现的次数，并对次数高的进行赋值。算法实例表明：新算法能减少求解过程中规则使用次数，减少求解步骤，尽早剪除不满足解空间，从而有效提高求解效率。 Aiming at the problem of the resolution efficiency of propositional satisfiability problem for propositional formula φ (CNF Form), based on the study of the DPLL complete algorithm and the research of the word order processing under different forms, a new algorithm for solving SAT problem is proposed. The algorithm groups propositional formula φ according to the length of clause, and sorts the words according to the number of polarity occurrence frequency in the initial sort. Then it considers the number of occurrences of the plus or minus words in variables and gets the value of high numbers. Algorithm instance shows that the new algorithm can reduce the number of rules in the process of using, reduce the solving steps, cut off the unsatisfy space of solution as soon as possible, so as to improve the solution efficiency.

SATenstein: Automatically building local search SAT solvers from components

Designing high-performance solvers for computationally hard problems is a difficult and often time-consuming task. Although such design problems are traditionally solved by the application of human expertise, we argue instead for the use of automatic methods. In this work, we consider the design of stochastic local search (SLS) solvers for the propositional satisfiability problem (SAT). We first introduce a generalized, highly parameterized solver framework, dubbed SATenstein, that includes components drawn from or inspired by existing high-performance SLS algorithms for SAT. The parameters of SATenstein determine which components are selected and how these components behave; they allow SATenstein to instantiate many high-performance solvers previously proposed in the literature, along with trillions of novel solver strategies. We used an automated algorithm configuration procedure to find instantiations of SATenstein that perform well on several well-known, challenging distributions of SAT instances. Our experiments show that SATenstein solvers achieved dramatic performance improvements as compared to the previous state of the art in SLS algorithms; for many benchmark distributions, our new solvers also significantly outperformed all automatically tuned variants of previous state-of-the-art algorithms.

Synchronous counting and computational algorithm design

Consider a complete communication network on n nodes. In synchronous 2-counting, the nodes receive a common clock pulse and they have to agree on which pulses are “odd” and which are “even”. Furthermore, the solution needs to be self-stabilising (reaching correct operation from any initial state) and tolerate f Byzantine failures (nodes that send arbitrary misinformation). Prior algorithms either require a source of random bits or a large number of states per node. In this work, we give fast state-optimal deterministic algorithms for the first non-trivial case f=1. To obtain these algorithms, we develop and evaluate two different techniques for algorithm synthesis. Both are based on casting the synthesis problem as a propositional satisfiability (SAT) problem; a direct encoding is efficient for synthesising time-optimal algorithms, while an approach based on counter-example guided abstraction refinement discovers non-optimal algorithms quickly.

Automating Boolean Set Operations in Mizar Proof Checking with the Aid of an External SAT Solver

In this paper we present the results of an experiment with employing an external SAT solver to strengthen the notion of obviousness of the Mizar proof checker. The presented extension of the Mizar system is based on a version of MiniSAT, called Logic2CNF. The SAT-enhanced Mizar checker is programmed to automatically spawn a new Logic2CNF process whenever it needs to justify any goal that can be solved by reducing it into a corresponding propositional satisfiability problem (equalities based on Boolean operations or set inclusion). The external tool is interfaced within the implementation of Mizar's requirements directives.

Set constraint model and automated encoding into SAT: application to the social golfer problem

On the one hand, constraint satisfaction problems allow one to expressively model problems. On the other hand, propositional satisfiability problem (SAT) solvers can handle huge SAT instances. We thus present a technique to expressively model set constraint problems and to encode them automatically into SAT instances. We apply our technique to the social golfer problem and we also use it to break symmetries of the problem. Our technique is simpler, more expressive, and less error-prone than direct modeling. The SAT instances that we automatically generate contain less clauses than improved direct instances such as in Triska and Musliu (Ann Oper Res 194(1):427–438, 2012), and with unit propagation they also contain less variables. Moreover, they are well-suited for SAT solvers and they are solved faster as shown when solving difficult instances of the social golfer problem.

Algorithms for computing backbones of propositional formulae

The problem of propositional satisfiability (SAT) has found a number of applications in both theoretical and practical computer science. In many applications, however, knowing a formula's satisfiab...